STRUCTURE OF CERTAIN PERIODIC RINGS AND NEAR-RINGS

Abstract

Using commutativity of rings satisfying (xy)n(x,y) = xy proved by Searcoid and MacHale [16], Ligh and Luh [13] have given a direct sum decomposition for rings with the mentioned condition. Further Bell and Ligh [9] sharpened the result and obtained a decomposition theorem for rings with the property xy =(xy)2f(x,y) wheref(X,Y)∈ Z < X,Y > , the ring of polynomials in two noncommuting indeterminates. In the present paper we continue the study and investigate structure of certain rings and near rings satisfying the following condition which is more general than the mentioned conditions : xy = p(x,y), where p(x,y) is an admissible polynomial in Z < X,Y >. Moreover we deduce the commutativity of such rings.